2008/03/10 by В. И. Богачев, Vladimir I. Bogachev, Alexander V. Kolesnikov +2
Mathematics · #35J60 #49Q20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities #math.DG #msc:35J60 #msc:49Q20
paper · pdf · doi:10.48550/arxiv.0803.1436
15 pages; minor changes
openalex publication_date 2008/03/10 · arxiv created 2008/05/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let A ⊂ ℝd, d≥ 2, be a compact convex set and let μ= \varrho0 dx be a probability measure on A equivalent to the restriction of Lebesgue measure. Let ν= \varrho1 dx be a probability measure on Br := \x\colon |x| ≤ r\ equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν= μ∘ T-1 and T = ϕ⋅ \rm n, where ϕ\colon A → [0,r] is a continuous potential with convex sub-level sets and \rm n is the Gauss map of the corresponding level sets of ϕ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth ϕ the level sets of ϕ are driven by the Gauss curvature flow x(s) = -sd-1 \frac\varrho1(s \rm n)\varrho0(x) K(x) ⋅ \rm n(x), where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions of the classical Gauss curvature flow starting from a convex hypersurface.